Analytic Geometry, Trig, Complex Numbers and Vectors (99-135)

99. Drawing to Scale with Coordinates  (grade 8 and 9):    The coordination of a figure in the plan (representing it by a set of coordinates) gives a means to draw it to scale (smaller, larger or same size). If S is a set of ordered pairs [a,b] and k is a scale factor (dilation constant), we may form the set  kS = { [ka,kb] |  [a,b] belongs to S}. Then we may declare the set of points in the plane corresponding to S and kS are similar. By examples  in the first instance (and by algebraic reasoning with aid of the Pythogorean distant formula and trigonometry later) we may observe that distance between corresponding points in S and kS are proportional, the proportionality constant being k, and that corresponding angles are equal.  The formally may be generalized to include translations and rotations of similar figures simple by accepting translations and rotations of attached coordinates systems.  All can be physically illustrated and implied by translations and rotation of figures and coordinate systems overlayed on maps and plans.  The approach here is informally, robust and extrinsic, without and apart from the limitations of pure mathematics and the restriction of the latters role to marks on paper.   Drawing to scale with coordinates implies all squares and  all circles are similar.  
    

    Numerical Exercises: Verify numerically that two circles with centered at the origin are similar  via a diliation. Verify numerically that two squares, side overlapping in part, with a common vertex at the origin are similar  via a dilation. Use SSS (or SAS or ASA) to draw a pair triangles with a common angle and vertex at origin.  Verify numerically for each method that triangles are similar via a dilation.

    A Slow Preamble Length Measurement Definition of Area Lengths and areas on maps and plans Maps drawn to scale Drawing to scale avoids distortions of angles and lengths Use of Maps not drawn to the same scale in all directions Use of Maps drawn to the same scale in all directions Use of Maps not drawn to scale, take II

    2. What is Similarity? Recognizing Like Shapes Similarity by design Similarity by Design Coordinate Viewpoint Similarity by Design Concept Codification In class shape drawing practices Coordinate View of Similarity Continued Translations, Rotations, Reflections, Dilatations Similarity of squares and circles Similarity of Triangles Similarity of Triangles- Application Similarity Calculations - Examples Similarity of Triangles- more

100. Difference of Coordinate Method, Absolutely |a-b|  (grade 9). Explain how and why the distance formula between two points on a coordinate axes, horizontal or vertical.  Take the absolute value of a real number (the difference) to be its unsigned part and as aside, show how calculate absolute value of x with a piecewise formula
     

     (**)	|x| = {	x	if x > 0
     		-x	if x < 0

     Note: The latter formula will be too algebraic for students if presented without a preamble.  Absolute value of a real number can be taken to be its unsigned part in the first instance.  The foregoing formulism (**) for |x| should appear after students have shown how to graph piecewise defined functions.
     
101. The Pythagorean Distance Formula  (grade 9):  Explain how and why this formula holds. Calculate distances between points in the same and different quadrants of the coordinate plane.  Show that the distance it gives for points on the same horizontal or vertical line agrees with the absolute difference of coordinate method. 
     
102. Similarity By Design (grades 9 and above):  Relation between lengths, areas and volumes of proportional figures in 1, 2 and 3D.
     
     The area of a region in the plane may be approximated by covering the region with a square grid, and counting the number of squares contained in the region. In the limit, as the grid or mesh size tends to zero, the total area of the counted squares should approach that of the region.  And if the limit exists, in a twist, we take its value to be the area of the region.  Now if lengths in the region and the square grids are scaled by a factor k,  an image of the region and the covering square results. The image squares have area k² times the area of the original squares, as easily explained.  But for any grid size, the number of squares covering (all in) the original region equals the number covering (or all in) the image region due to a one to one correspondence. It follows that the approximations and the area of the image region (a limit of the approximations) are k² times the corresponding approximation and area of the original region.  In general, that implies when a figure appears twice, one at k times the length scale of the other, the area of the image figure will be  that of the original, and the length of any curve in the image of the figure will be k times that of corresponding preimage curve.  
     
     In general, if unit lengths are scaled by a factor k then unit areas are scaled by factor  k² and unit volumes are scaled by a factor k³ .  Now if any length, area or volume of a object is taken to be the limit of a the length, area or volume approximation by covering by intervals, squares or cubes then the same number of intervals, areas or cubes (scaled) with cover or approximate the length, areas or volumes. It follows that the corresponding lengths, areas and volume approximation alone and in the limit scale by the factors k, k²  and k³, respectively.
     
103. Midpoint Formula for a pair of points  [x₁, y₁] and [x₂, y₂] (grade 9 or 10).   First Technical Exercise: For multiply examples, plot the points
     
     f(t) = [x₁, y₁]+ t[ x₂ - x₁, y₂-y ₁]   (linear interpolation formula)
     
     and observe it provide a straight segment.  Observe or Show the distance of f(t) to the initial endpoint  is t times the distance between the two end points. 
     
     Conclusion:  f(½) = [½ (x₁+ x₂ ) , ½ (y₁+ y₂) ] is the midpoint of the line segment joining [x₁, y₁] and [x₂, y₂]
     
104. Dilatation Exercise (grade 9 or 10):   the distance of [kx,ky] to the origin [0,0] is k times the distance of [x,y] to the origin.   For [x,y] nonzero, plotting [kx,ky] for various values of k suggests all points belong a line through [x,y] and [0,0], and each point on that line corresponds to exactly one value of k. 
     
105. Polar Coordinates (grade 8 or 9). Learn how to measure polar coordinates of given nonzero point [x, y]  in the plane, and given the polar coordinates for a point, learn how to locate the  points and to determine or measure it rectangular coordinates [x,y].   Get familiar with the polar coordinates (modulo 360 degrees) of many points in the plane on the coordinate axes and inside all four quadrants.  
     
106. Complex Numbers (grade 8 or 9).  Introduce points in the plane as complex numbers.  learn about their rectangular and polar form or coordinates for these "numbers". Say how add them using rectangular coordinates (form). Learn to multiply using the polar form. It follows immediately that sums and products may be calculated using subtotals and subproducts.  The aforementioned geometric view of products of unsigned numbers and the polar form introduction of multiplication for points in the plane implies the product of nonzero complex numbers is nonzero.  
     
107. Rigid Body Movement (grade 8 or  9).    Observe if a triangle with one vertex at the origin is rotated about the origin, then the midpoint of the opposite side remains the midpoint under a rotation. So rotations commute with midpoint calculations, and so rotations distribute over midpoint calculations. 
     
108. Distributive Law for Complex Numbers (grade   9). Now pre-multiplication of sum of two complex numbers by a complex numbers may be cast as the midpoint of the complex numbers being subject to a rotation and a dilatation. As rotation and dilatation distribute over midpoint calculations it follows that complex number multiplication is distributive.   The expression of complex numbers in terms of real and imaginary parts means the distributive law for complex numbers is inherited by real numbers, and also implies complex number operations on real numbers coincide with those on real numbers.
     
109. Three or Six periodic trig functions (grade 9 or 10). Introduce  with the aid of points in the  unit circle.  Use the complex number valued function cis(t) = cos(t) + i sin (t)  and the algebraic properties of complex numbers to imply Pythagorean trig identity, reciprocal identities, angle sum formulas for sine and cosine, double angle formulas and half-angle formulas. Derive trig formulas for dot and cross-products.  For lines y = m x through the origin, introduce their slope m as the tangent of an angle of inclination.  
     
110. Trig Functions and Right Triangles (grade 9 or 10).  For a right triangle, select a coordinate system with  that its legs are parallel to the coordinate axes,  hypotenuse is a line segment in the first quadrant with one end at the origin. Use the polar form of the coordinate of the non-origin end of the hypotenuse to express trig function as ratios of side lengths - the hypotenuse being one side. Next use those trig ratios forwards and backwards to solve for missing lengths and angles in right triangles.  
     
     Note: As part of the six application areas recommended for primary and junior high school mathematics, studies with drawing maps, plans and figures to scale, should have early emphasized the latter as a practical  means to finding missing angles and lengths. With that experience, often missing today, students may view the trig methods for solving triangles alone and adjacent to each other in complicated problems as alternative route, a refinement in which sketches may serve in place of carefully drawn to scale diagrams.   
     
     In the approach above, we have reversed earlier development of six trig functions in which the definition of them in terms of ratios of sides of right triangle required the discussion of proportionality of sides in  similarity right triangles to imply trig functions were well defined - one nuance too many for many students.   Starting with the unit circle view obviates the need for that discussion, but uses the similarity of right triangles  to apply the unit circle defined trig functions to right triangles. (The nuance that the definition of trig functions is independent of the size of the unit length can be left to a course on pure mathematics - an application of similarity theory.)
     
111. Values of 3 Trig Functions at 45, 30 and 60 degrees (grade 9 or 10).  Show how to use isosceles right triangle (legs of length 1) and how to bisect the equilateral triangle, sides of length 2, to find these values. 
     
112. Values of Trig Functions for 3-4-5 and 5-12-13 right triangles (grade 9 or 10).  Use inverse trig functions to find the missing angles in these triangles, and verify with a protractor. Also compare measured values with those provided by inverse trig functions for sine, cosine and tangent.   The redundancy here may be cast as informative.
     
113. Sine laws for Triangles (grade 9 or 10).  Derive and explain what happens or simplifies in the case of right triangles.   The derivation may include three cases:  the perpendicular from a vertex opposite a side selected to serve as a base of the triangle falls (i) inside the base, (ii) to one side of the base, and (iii) on an end point of the base. Talk about the forward and backward use of simultaneous sine laws.  Note the introduction of trig functions with the aid of the unit circle implies sine is defined for acute and obtuse angles. 
     
     Note the details of the derivation in essence provide another example of the forward and backward use of the sine function - the exercise prove or reproduce proofs of the sine could be a set as problem.  In trig, anything exercise to show that something holds may framed as a proof.  So some proofs provide experience or models to follow in solving problems. 
     
114. Cosine laws (grade 9 or 10).  Derive and explain what happens or simplifies in the case of right triangles.   The derivation here may use properties of complex numbers if a coordinate system is located at the vertex where the angle used in the cosine law. Talk about the forward and backward use. Talk  about which one to use and when.
     
115. Reference Angles (grade 9 or 10). Use of 45, 30 and 60 degrees as Reference Angles for themselves plus multiples of 180 degrees.  These angles are associated with special triangles: isosceles right triangle and half an equilateral triangle - one with sides of length 2 or 1. 
     
116. Half-Angle Formula (grades 9 or 10) .  Derive  Then Calculate values trig functions at 15 degrees and 22.5 degrees.
     
117. Trigonometric Identities (grades 9 or 10)   Make this topic simpler (reduce to an algebraic exercise) by using Euler complex number formulas for sine and cosine to simplify some questions.  Given mastery of complex numbers, there is a question of to what extent trig identities should be developed and explored. The algebraic approach provided by complex number properties simplifies everything.  The complex number approach appears in undergraduate courses in engineering and science. That begs the question of why the high school development should follow the harder route.
     
118. Linear Functions (grades 9 or 10):  Draw the graph of a linear function y = mx + b.  Show numerically and algebraically that for pairs of points [x₁,y₁] and [x₂,y₂] in the graph that m = (y₂-y₁)/(x₂-x₁). Recognize the slope m as a proportionality constant for the rise over run ratio. Also show m = the tangent of the angle of inclination of the line with any intersecting horizontal line. Algebraically derive and numerically apply point-slope, two point, and slope intersect equations for the line. Use methods for solving systems of two equations in two unknowns to find intersection points of lines with unequal slope.  Understand why the product of  slopes of perpendicular lines (neither vertical) is -1.  Recognize slope as a proportionality constant when b = 0 - the case of a line through the origin.  Recognize slope as a rate of change when y and x are given by quantities (numerical multiples of a unit of measure), and observe in the case where y = distance and x = time that m = speed. 
     
119. Composition and Inverses of Linear Functions (grades 9 or 10)  (i) If f(x) = ax+ b and g(x) = cx+d then h(x) = f(g(x)) has slope ac. That is the precalculus version of the chain rule for composition or substitution of one linear function into another.  Students may find this easier to understand if we say  r = as+ b = f(s) and s = cp+ d = g(q)  then r = as + b = a(cp+d) + s = acp+ (ad+s) is a linear function of p with slope or rate of change ac.  (ii) If f(x) = ax+ b and g(x) = cx+d are an inverse pair of functions so that  f(g(x)) = x then the product ac of their slopes is 1. The slope of each is the reciprocal of the other. 
     
     
120. A Why Slopes Calculus Preview (grades 9 or 10).  Explain that at each point on the graph of a sufficiently smooth curve y = f(x), that there is or should be a slope. Explain that calculus largely consists of the direct and reverse calculation of slopes to nonlinear functions y = f(x).  
     
     Explain that calculus is the part of mathematics that provides an efficient language for the description of rates and further concepts in accounting, engineering, science and technology.  Explain that most high school courses represent preparation for calculus, the mathematical subject that employs most earlier skills and know-how at full strength - forgetfulness not allowed. 
     
121. Quadratics (grade 10 and above). First show how zeroes of quadratic with real coefficients may be found when they are given  as a product of two linear factors. Next show how to factor the difference of two squares and apply that to factor some quadratics. Next show how to complete the square to express quadratics as the difference of two squares with or without the aid of the complex number i = square root of -1.  Then factor.   Next. show how to do the foregoing algebraically in order to derive the quadratic formula.
     
122. Graphing and Applying Quadratics Functions (grade 10 and above):  For  quadratics with real coefficients, observe how completing the square essentially leads to a constant times  sum of two squares,  a constant times the difference of two squares or a constant times a  square.  The process may be described algebraically. In any event, for quadratics f(x) = ax²+bx+c, the process leads to a formula f(x) =  a[ (x-h)² + k]   where k is positive, zero or negative.  In the first case, the quadratic has no real roots and its graph lies on one side of the horizontal axes. In the second case, the quadratic has two real roots and its graph intersects the horizontal axes at two points. In the last case, the quadratic touches the horizontal axes at one points, and the rest of it lies on one side.   Use ability to solve quadratic equations in solving fractional equations. Use the quadratic formula and the form f(x) =  a[ (x-h)² + k] forwards and backwards to graph quadratics and to determine or interpret coefficients: a, b, c, h and k etc.  Show how coefficient may shift the graph horizontally and vertical, and rescale the range (y) values.  Show how to compress a finite portion of the graph horizontally.
     
123. Physical Application of Quadratics (grade 10 and above):  Show that displacement of the form x = x_o + vt model constant velocity movement.  With the aid of algebraic and numerical limit evaluation, show that the vertical movement  y = y_o + v_ot + ½ at² has velocity  v =   v_ot + at and acceleration a.    More motion examples may be employed and studied here forwards and backwards.
     
     Note: The algebraic description of limits may be employed here. 
     
124. Solve Systems of two equations, one linear and one quadratic (Grades 10 and above).   Learn how to solve and apply to projectile problems - the question of where a projectile traveling in a vertical lands on an incline for example. 
     
125. Factor Sums and Difference of two cubes (grade 10 and above):  Verify using multiplication and use forwards and backward. 
     
126. N-th Roots of Unity and Complex Numbers in General (grade 10 and above):  Factor t² -1,   t³ -1, t⁴ -1,t⁵ -1 and t⁶ -1 and plot their roots around the unit circle.   Explain and/or derive De Moivre's formula.   Regular Polygons.
     
     
127. Fundamental Theorem of Algebra (grade 10 and above):: The Fundamental theorem of Algebra says any polynomial of degree n with complex coefficients has n complex roots.  From that, long division of polynomials implies each polynomial of degree can be written as a constant times a product of n linear factors of the form (x-z_m) where for 1 < m < n, z_m is complex number. In the case of polynomials with real coefficients, the complex roots occur in complex conjugate pairs.  It follows that each polynomial of degree n with real coefficients is a constant times a product of linear factors, each corresponding to a real root, times a product of quadratic factors, each having  and  a conjugate complex pair of roots, roots inherited from the original polynomials. In this revisit previous examples of factored polynomials. 
     
128. Why Factor Polynomials (grade 10 and above)::  This second algebraic preview of the role of slopes in calculus may provide a context for this and also develop algebraic sense or reasoning skills. The zeroes of factors in polynomials and rational functions locate extreme points and asymptotes. 
     
129. Differential Calculus With Polynomials:  Introduce algebraic development of limits for polynomials. Apply to find the derivatives of constant, linear and quadratic functions.  Then develop general formulas for polynomials by induction.  
     
130. Radian Measure (grade 11 and above)::  For the arc of a circle, show that the pure number arclength divided by radius is proportional to the angle subtended by the arc. So knowledge of one determines the other. Then express multiples of 15 degrees in terms of radian measure the value of arclength divided by radius.  Explain that the angular unit of measure, the radian, has the numerical value 1 and that appearance is a cosmetic convenience - employed to indicate that number is gives measure of an angle in radians.   Immediate Application: In physics course, describe constant speed motion around a circle.  Future Application: Explain that in calculus that formulas for slopes to the sine and cosine curve become simpler with the use of radian measure.  Review values of sine, cosine and tangent functions at multiples of 30 degrees ( p/6) and multiples of 45 degrees ( ¼p).
     
131. Vectors in Plane (grade 10 and above):  Introduce vectors as ordered pairs of numbers which determine a displacement or movement.  Explain how the be drawn or illustrated with head and tail with tail located at point in the plane. Show how vectors drawn at the same spot may be added with  coordinates. Show how that determines a parallelogram, and how head to tail addition provides an alternative to the coordinate description of the addition of vectors.  Introduce dot and cross products.  Recall how previous mastery of complex numbers implies trig formulas for the value and magnitude of the latter. Show how to multiple by real numbers and so form scalar multiples.  Determine magnitude of product.  Show or recall how vectors may be used in sequence (head to tail) to plan or draw a piecewise linear route on a map. Mention the use of vectors in physics to represent velocity, acceleration, forces and fields: electric, magnetic, etc.  
     
     Applications: Vectors appear in the representation of displacements on maps, in physics in the the modeling of velocity, acceleration, momentum and force, and fields. Vectorial ideas may also appear in the explanation or development of operations with signed numbers. 
     
     Aside: The interpretation of dot products in terms of the angle between two vectors rests on the identification of a orthogonal coordinate system x'y'z' in which the vectors are horizontal (have no z'-component), on the invariance of the form the dot product under an orthogonal changes of variables in general and hence in the particular case of the change from the original xyz- to the x'y'z' coordinate system.  The discussion of conic section is related to this matter. 
     
     
132. Three Inverse Trig Functions (grade 11 and above): Graph sine, cosine and tangent versus angles with the latter expressed in terms of radian measure.  Show how to restrict the domain of each, so that the horizontal line method for calculating a value from a set of ordered pairs provide an inverse function to the restricted domain versions of sine, cosine and tangent.  Show how to evaluate the inverse functions at special values, those corresponding to their values at reference angles etc. The foregoing may formalize (provide theory) to sanction earlier practices in finding missing angles in right triangles. 
     
133. Sinusoidal Functions Forwards and Backwards (grade 11 and above):  Show how to graph y = A sin(wt+c) and y = A cos(vt+d), how the graphs of these functions are related, and how to express sums  Csin(wt) + D cos(wt) in terms of the form. Talk about amplitude and phase shift. Relate to the study of waves and motions in physics.    
     
134. Functions: (grade11 and above, replaceable).  Modern mathematics high school  curricula identifies real value functions y  = f(x) of a real variable x with sets of order pairs in the plane with the vertical line property. Relations too are identified with sets of orders.  The full set theory view theory of such Functions may be introduced after mastery of  polynomial functions (linear and quadratic and cubic included), after the forwards and backward discussion of logarithms and exponentials, and after the forward and backward discussion of three to six trigonometric functions and their inverses. If these examples are put first, then the general theory of functions may stand on them and represent a summary or small refinement of concepts and operations previously met.  In other words function skills and concepts instead of being present all at once, may be unfolded in context as needed for each of the common functions, those just mentioned and possible more, one at a time, one after another.   Food for Thought: The full theory may be part of the structure of modern mathematics, but its presence in secondary mathematics may not be required.     
     
135. Conic Sections (grade 11 and above):  From loci description, derive formulas for ellipses, parabolas and hyperbolas in the standard  cases with axes parallel to a coordinate axis.   Use translation to move the latter in plane. Use completing the square to convert quadratic equations in two variable x and y say, with no xy terms, into standard form for quadratics equations associated with axes parallel to a coordinate axes. 
     
     Note:  The intersection of a plane with a cone is an ellipse, parabola or hyperbola.   To show this, let the axis of symmetry of a cone provide the z-axis of a coordinate system with the cone vertex located at the origin.  The normal to a plane  and the axis of symmetry of the cone determine a xz-plane with the z-axis as just described, and the vertex at the origin of this xz plane.  Let the y-axis be through the vertex and perpendicular to the xz plane.  In the foregoing selected coordinate system, the cone has equation  z² = p(x²+y²) where p = tan(b) where  0 < b < 90 degrees,  and the plane has equation z = mx+ q where m = tan (a).  Rewriting the as   cos(a) x - sin(a) z = k allows the limiting case  a = 90 degrees - the case of intersection with a vertical plane x = k.  Now
     

     x'  =

     mx        sqrt(1+m2)

     and  y' = y

     provide an orthogonal coordinate system in the plane cos(a) x - sin(a) z = k.  The solution of the system of equations
     
       z² = tan(b) (x²+y²)   and   cos(a) x - sin(a) z = k

     via elimination of z leads to equations in standard form for ellipses, parabolas and hyperbolas in generic and limiting cases corresponding  in three cases. Very quickly,  I suspect that case (i) |a| < b gives a circle when  a = 0 and an ellipse otherwise; case (ii) |a| = b leads to parabola; and case (iii) |a| > b gives a hyperbola, or in the limiting case k = 0, a pair of intersecting lines.   .
     
     Exercise: Express the eccentricity of each conic section in terms of k and the angles  a  and  b.
     
     Point of Investigation  - a site to do: Find an old textbooks which show how why the intersection of a cone with a plane leads to loci of points corresponding to an parabola, ellipse or hyperbola. 
     
     Further Note: In physics, orbital motion of a mass around another is modeled by conic sections. In mechanics again, quadratics appear in the study of finite and "infinite" matrices and their properties. In calculus of functions of two variables, level sets of quadratics (those association with Hessian matrices) are used in the identification of high points, low points and saddle points.  A year of calculus for functions of one variable may come before the study of calculus for two and more variables. So the study of quadratics and conics might be left to course on matrices and linear algebra.  The diagonalization of the symmetric (Hessian) matrix for a quadratic by an orthogonal change of coordinate, based perhaps on completing the square or matrix methods, implies the level set of a quadratic will be an ellipse, parabola or hyperbola, except in degenerate (limiting) cases. 

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science